Question: Solve for $x$ : $ 4|x - 10| - 5 = 1|x - 10| + 6 $
Explanation: Subtract $ {1|x - 10|} $ from both sides: $ \begin{eqnarray} 4|x - 10| - 5 &=& 1|x - 10| + 6 \\ \\ { - 1|x - 10|} && { - 1|x - 10|} \\ \\ 3|x - 10| - 5 &=& 6 \end{eqnarray} $ Add ${5}$ to both sides: $ \begin{eqnarray} 3|x - 10| - 5 &=& 6 \\ \\ { + 5} &=& { + 5} \\ \\ 3|x - 10| &=& 11 \end{eqnarray} $ Divide both sides by ${3}$ $ \dfrac{3|x - 10|} {{3}} = \dfrac{11} {{3}} $ Simplify: $ |x - 10| = \dfrac{11}{3}$ Because the absolute value of an expression is its distance from zero, it has two solutions, one negative and one positive: $ x - 10 = -\dfrac{11}{3} $ or $ x - 10 = \dfrac{11}{3} $ Solve for the solution where $x - 10$ is negative: $ x - 10 = -\dfrac{11}{3} $ Add ${10}$ to both sides: $ \begin{eqnarray} x - 10 &=& -\dfrac{11}{3} \\ \\ {+ 10} && {+ 10} \\ \\ x &=& -\dfrac{11}{3} + 10 \end{eqnarray} $ Change the ${ + 10}$ to an equivalent fraction with a denominator of $3$ $ x = - \dfrac{11}{3} {+ \dfrac{30}{3}} $ $ x = \dfrac{19}{3} $ Then calculate the solution where $x - 10$ is positive: $ x - 10 = \dfrac{11}{3} $ Add ${10}$ to both sides: $ \begin{eqnarray} x - 10 &=& \dfrac{11}{3} \\ \\ {+ 10} && {+ 10} \\ \\ x &=& \dfrac{11}{3} + 10 \end{eqnarray} $ Change the ${ + 10}$ to an equivalent fraction with a denominator of $3$ $ x = \dfrac{11}{3} {+ \dfrac{30}{3}} $ $ x = \dfrac{41}{3} $ Thus, the correct answer is $x = \dfrac{19}{3} $ or $x = \dfrac{41}{3} $.